Journal of Statistical Planning and Inference 137 (2007) 3352 – 3360 www.elsevier.com/locate/jspi A smooth version of the step-up procedure for multiple tests of hypotheses Arthur Cohen ∗, 1 , John Kolassa 2 , Harold B. Sackrowitz 1 Department of Statistics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA Available online 31 March 2007 Abstract Cohen and Sackrowitz [2005. Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure. Ann. Statist. 33, 145–158; 2007. More on the inadmissibility of step-up. J. Multivariate Anal. 97, 481–492] have demonstrated that the popular step-up (SU) multiple testing procedure is inadmissible under a wide variety of conditions. All conditions, however, did assume a permutation invariant (symmetric) model. In this paper we find a necessary condition for admissibility of multiple testing procedures in the asymmetric case. Once again SU does not satisfy the condition and is inadmissible. Since SU has a somewhat less favorable practical property and a less favorable theoretical property, we offer a smooth version of SU which retains the favorable practical properties and avoids some of the less favorable ones. In terms of performance the smooth version and nonsmooth version seem to be comparable at least in low dimensions. © 2007 Elsevier B.V.All rights reserved. MSC: 62F03; 62C15 Keywords: Admissibility; Permutation invariant procedures; Nonsymmetric procedures; Exponential family; Classification risk function 1. Introduction Multiple testing procedures are enjoying a resurgence of interest as a result of new applications to microarrays, stock market mutual funds, educational testing, clinical trials, and psychological experiments. One of the most popular approaches is the step-up (SU) procedure. The version put forward by Benjamini and Hochberg (1995) is designed to control the false discovery rate (FDR). This version has received considerable attention. See, for example, Efron (2003), Genovese and Wasserman (2002), Sarkar (2002) and Dudoit et al. (2003). The latter reference surveys other methods and lists 18 step-wise procedures, six of which are SU. Step-wise procedures have an advantage over Bonferroni-type single step procedures in that they have better power in some sense. They also have more flexibility than a single-step procedure in the sense that all the data is used when testing each individual hypothesis. Nevertheless, Cohen and Sackrowitz (CS) (2005, 2007) have shown that SU procedures are inadmissible under a wide variety of conditions. The variety of conditions include different distributions, many dependent situations, two- sided and one-sided alternatives and different risk functions. All the cases treated by CS thus far involve a permutation ∗ Corresponding author. Tel.: +1 7324455305; fax: +1 7324453428. E-mail address: artcohen@rci.rutgers.edu (A. Cohen). 1 Research supported by NSF Grant DMS-0457248 and NSA Grant H98230-06-1-007. 2 Research supported by NSF Grant DMS-0505499. 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.03.016